[Paper Review] Pathwise Taylor Expansions for Itô Random Fields

Rainer Buckdahn, Ingo Bulla+1 more|arXiv (Cornell University)|Aug 19, 2010

Stochastic processes and financial applications19 references3 citations

TL;DR

This paper introduces a universal pathwise stochastic Taylor expansion for Itô-type random fields with self-exciting diffusion, where the diffusion depends on both the solution and its spatial derivatives. The expansion is valid around arbitrary random time-space points $(\tau,\xi)$, with a null set independent of the choice of $(\tau,\xi)$, enabling a new definition of stochastic viscosity solutions for fully nonlinear SPDEs with gradient-dependent diffusion coefficients.

ABSTRACT

In this paper we study the {\it pathwise stochastic Taylor expansion}, in the sense of our previous work \cite{Buckdahn_Ma_02}, for a class of Itô-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an "self-exciting" type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of "$n$-fold" derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work \cite{Buckdahn_Ma_02}, our new expansion can be defined around any random time-space point $( ,ξ)$, where the temporal component $ $ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point $( ,ξ)$. As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the {\it stochastic viscosity solution} for such SPDEs, which is new in the literature.

Motivation & Objective

• To extend pathwise stochastic Taylor expansions to Itô random fields whose diffusion depends on both the solution and its spatial derivatives.
• To remove the restriction that the expansion point $(\tau,\xi)$ must be a stopping time, allowing arbitrary random points.
• To ensure the exceptional null set in the expansion is independent of the choice of $(\tau,\xi)$, enhancing universality and applicability.
• To apply the new expansion to define and analyze stochastic viscosity solutions for fully nonlinear SPDEs with gradient-dependent diffusion.
• To overcome limitations of previous Doss-Sussmann transformations in anticipating stochastic calculus settings.

Proposed method

• Introduce the notion of $n$-fold derivatives of a random field to handle the self-exciting nature of the diffusion term.
• Construct a pathwise Taylor expansion around arbitrary random points $(\tau,\xi)$, not necessarily stopping times.
• Use a universal null set across all random points, ensuring the expansion holds almost surely for all such points.
• Define a process $\phi_t$ via a stochastic flow associated with the test field $\varphi$, enabling evaluation along stochastic characteristics.
• Apply the expansion to $u(t, \phi_t(x, D\psi(t,x)))$ and use the resulting expansion to derive conditions for viscosity subsolution properties.
• Establish that the solution satisfies the stochastic viscosity subsolution condition by comparing $u$ and $\varphi$ at the random point $(\tau,\xi)$, leveraging the expansion's remainder terms.

Experimental results

Research questions

• RQ1 Can a pathwise stochastic Taylor expansion be constructed for Itô random fields whose diffusion depends on both the solution and its spatial derivatives?
• RQ2 Does the expansion remain valid when the expansion point $(\tau,\xi)$ is an arbitrary random time-space pair, not necessarily a stopping time?
• RQ3 Is it possible to construct a single null set that works for all random expansion points, ensuring universality of the expansion?
• RQ4 How can such a universal expansion be used to define and analyze stochastic viscosity solutions for fully nonlinear SPDEs with gradient-dependent diffusion?
• RQ5 Can the new expansion overcome the limitations of the Doss-Sussmann transformation in anticipating stochastic calculus?

Key findings

• The paper establishes a universal pathwise stochastic Taylor expansion for Itô random fields with self-exciting diffusion, valid around any random time-space point $(\tau,\xi)$, with a null set independent of the choice of $(\tau,\xi)$.
• The expansion is constructed using $n$-fold derivatives of the random field, enabling handling of the recursive, self-exciting structure in the diffusion coefficient.
• The method allows the definition of stochastic viscosity solutions for fully nonlinear SPDEs whose diffusion depends on both the solution and its gradient, a new contribution to the literature.
• The expansion is used to prove that a classical solution $u$ is a stochastic viscosity subsolution by showing $f(\xi, (u,Du,D^2u)(\tau,\xi)) - \theta(\tau,\xi) \geq 0$ almost surely on the left-maximum set.
• The same argument applies to supersolution, proving that $u$ is a stochastic viscosity solution, thus establishing the new solution concept for gradient-dependent SPDEs.
• The approach avoids reliance on the Doss-Sussmann transformation, which fails in the self-exciting case, and instead uses a novel pathwise expansion with universal null sets.